NONUNIQUENESS IN INVERSE ACOUSTIC SCATTERING ON THE LINE

The generalized one-dimensional Schrodinger equation d2phi/dx2+k2H(X)2phi =P(x)phi is considered. The nonuniqueness is studied in the recovery of the function P(x) when the scattering matrix, H(x), and the bound state energies and norming constants are known. It is shown that when the reflection coefficient is unity at zero energy, there is a one-parameter family of functions P(x) corresponding to the same scattering data. An explicitly solved example is provided. The construction of H(x) from the scattering data is also discussed when H(x) is piecewise continuous, and two explicitly solved examples are given with H(x) containing a jump discontinuity.